Connectivity of path graphs

Abstract

In the paper we present lower bounds for the connectivity of a path graph P 2 (G) of a graph G. Let δ ≥ 3 be the minimum degree of G. We prove that if G is a connected graph, then P 2 (G) is at least (δ−1)connected; and if G is 2-connected, then P 2 (G) is at least (2δ−2)connected. We remark that if G is a δ-regular graph then P 2 (G) is (2δ−2)-regular, and hence, if G is 2-connected then P 2 (G) is (2δ−2)connected; its theoretical maximum.

Edge-connectivity in Pk-path graphs

“…We also find recent results on vertex connectivity of iterated line graphs in [10]. The vertex-connectivity of P 2 -path graphs has been studied in [11] and [13]. As a direct consequence of the result shown in [8], it follows that if G is a connected graph with at most one vertex of degree one, then P 2 (G) is also connected.…”

“…The vertex-connectivity of P 2 -path graphs has been studied in [11] and [13]. As a direct consequence of the result shown in [8], it follows that if G is a connected graph with at most one vertex of degree one, then P 2 (G) is also connected. In [2] the following theorem is derived.…”

“…In [8], there is a necessary and sufficient condition for a P k (G)-path graph to be disconnected assuming G has girth at least k + 1. As the condition implies that G must have vertices of degree 1, it follows that if G has girth g(G) k + 1 and minimum degree (G) 2, then P k (G) is connected.…”

A sufficient condition for Pk-path graphs being r-connected

“…Recent results on vertex-connectivity of iterated line graphs are provided by Knor and Niepel [12]. The vertex-connectivity of P 2 -path graphs has been studied by Knor et al [13] and by Li [15]. From a result showed in [11], it is not difficult to see that if G is a connected graph with at most one vertex of degree one, then P 2 (G) is also connected.…”

“…In [2] the edge-connectivity of P 2 -graphs is studied giving lower bounds on the edge-connectivity which are expressed in terms of the edge-connectivity of G. These latter bounds are generalized in [3] for k 2. Recently, Knor and Niepel [10] have studied the connectivity of P 3 -path graphs, and furthermore the following sufficient condition to guarantee connected P k -path graphs for k 2 is easily derived.…”